Multi-dimensional codes are widely used due to their potential efficiency. In NAND flash memory systems the reliability requirement dictates operation in extremely low output bit error rate (BER), for example, this can be output BER that is below ten by the power of minus fifteen.
For achieving such goal the encoding system should have a very low error-floor. There are various methods of designing codes and decoder for low error floor. Novel methods for encoding and decoding for obtaining low-error-floor codes are suggested in this application.